ch04_06 - 4-51 SECTION 4.6 2. When solving differential...

4-51 SECTION 4.6
. .
Integration by Substitution 393
2. When solving differential equations of the form
dy
dt
= f ( y )
for the unknown function y ( t ), it is often convenient to make
use of a potential function V ( y ). This is a function such that
−
dV
dy
= f ( y ). For the function f ( y ) = y − y 3 , find a potential
function V ( y ). Find the locations of the local minima of V ( y )
and use a graph of V ( y ) to explain why this is called a “double-
well” potential. Explain each step in the calculation
Since dV
dt
≤ 0, does the function V increase or decrease as
time goes on? Use your graph of V to predict the possible val-
ues of lim
t →∞
y ( t ). Thus, you can predict the limiting value of
the solution of the differential equation without ever solving
the equation itself. Use this technique to predict lim
t →∞
y ( t ) if
y = 2 − 2 y .
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EXAMPLE 6.1 Finding an Antiderivative by Trial and Error
Evaluate 2 xe
x 2
dx .

394 CHAPTER 4
. .
Integration 4-52
Note that, in general, if F is any antiderivative of f , then from the chain rule, we
have
d
dx
[ F ( u )] = F ( u )
du
dx
= f ( u )
du
dx
.
From this, we have that
f ( u )
du
dx
dx =
d
dx
[ F ( u )] dx = F ( u ) + c = f ( u ) du , (6.1)
since F is an antiderivative of f . If you read the expressions on the far left and the far right
sides of (6.1), this suggests that
du =
du
dx
dx .